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Interatomic Bonding in Solids: Fundamentals, Simulation, and Applications

Interatomic Bonding in Solids: Fundamentals, Simulation, and Applications

Valim Levitin

ISBN: 978-3-527-67155-7

Mar 2014

320 pages

Description

The connection between the quantum behavior of the structure elements of a substance and the parameters that determine the macroscopic behavior of materials has a major influence on the properties exhibited by different solids. Although quantum engineering and theory should complement each other, this is not always the case.

This book aims to demonstrate how the properties of materials can be derived and predicted from the features of their structural elements, generally electrons. In a sense, electronic structure forms the glue holding solids together and it is central to determining structural, mechanical, chemical, electrical, magnetic, and vibrational properties. The main part of the book is devoted to an overview of the fundamentals of density functional theory and its applications to computational solid-state physics and chemistry.

The author shows the technique for construction of models and the computer simulation methods in detail. He considers fundamentals of physical and chemical interatomic bonding in solids and analyzes the predicted theoretical outcome in comparison with experimental data. He applies first-principle simulation methods to predict the properties of transition metals, semiconductors, oxides, solid solutions, and molecular and ionic crystals. Uniquely, he presents novel theories of creep and fatigue that help to anticipate, and prevent, possibly fatal material failures.

As a result, readers gain the knowledge and tools to simulate material properties and design materials with desired characteristics. Due to the interdisciplinary nature of the book, it is suitable for a variety of markets from students to engineers and researchers.

Preface XI

1 Introduction 1

2 From Classical Bodies to Microscopic Particles 7

2.1 Concepts of Quantum Physics 7

2.2 Wave Motion 11

2.3 Wave Function 13

2.4 The SchrödingerWave Equation 14

2.5 An Electron in a SquareWell: One-Dimensional Case 16

2.6 Electron in a Potential Rectangular Box: k-Space 18

3 Electrons in Atoms 21

3.1 Atomic Units 21

3.2 One-Electron Atom: Quantum Numbers 22

3.3 Multi-Electron Atoms 30

3.4 The Hartree Theory 33

3.5 Results of the Hartree Theory 35

3.6 The Hartree–Fock Approximation 39

3.7 Multi-Electron Atoms in the Mendeleev Periodic Table 40

3.8 Diatomic Molecules 43

4 The Crystal Lattice 49

4.1 Close-Packed Structures 49

4.2 Some Examples of Crystal Structures 50

4.3 The Wigner–Seitz Cell 53

4.4 Reciprocal Lattice 54

4.5 The Brillouin Zone 59

5 Homogeneous Electron Gas and Simple Metals 63

5.1 Gas of Free Electrons 64

5.2 Parameters of the Free-Electron Gas 66

5.3 Notions Related to the Electron Gas 69

5.4 Bulk Modulus 69

5.5 Energy of Electrons 70

5.6 Exchange Energy and Correlation Energy 71

5.7 Low-Density Electron Gas: Wigner Lattice 74

5.8 Near-Free Electron Approximation: Pseudopotentials 74

5.9 Cohesive Energy of Simple Metals 77

6 Electrons in Crystals and the Bloch Waves in Crystals 79

6.1 The Bloch Waves 79

6.2 The One-Dimensional Kronig–Penney Model 82

6.3 Band Theory 85

6.4 General Band Structure: Energy Gaps 87

6.5 Conductors, Semiconductors, and Insulators 91

6.6 Classes of Solids 92

7 Criteria of Strength of Interatomic Bonding 95

7.1 Elastic Constants 95

7.2 Volume and Pressure as Fundamental Variables: Bulk Modulus 98

7.3 Amplitude of Lattice Vibration 98

7.4 The Debye Temperature 102

7.5 Melting Temperature 102

7.6 Cohesive Energy 103

7.7 Energy of Vacancy Formation and Surface Energy 105

7.8 The Stress–Strain Properties in Engineering 106

8 Simulation of Solids Starting from the First Principles (“ab initio” Models) 109

8.1 Many-Body Problem: Fundamentals 109

8.2 Milestones in Solution of the Many-Body Problem 112

8.3 More of the Hartree and Hartree–Fock Approximations 112

8.4 Density Functional Theory 115

8.5 The Kohn–Sham Auxiliary System of Equations 118

8.6 Exchange-Correlation Functional 119

8.7 Plane Wave Pseudopotential Method 121

8.8 Iterative Minimization Technique for Total Energy Calculations 124

8.9 Linearized Augmented PlaneWave Method 127

9 First-Principle Simulation in Materials Science 131

9.1 Strength Characteristics of Solids 131

9.2 Energy of Vacancy Formation 134

9.3 Density of States 135

9.4 Properties of Intermetallic Compounds 136

9.5 Structure, Electron Bands, and Superconductivity of MgB2 138

9.6 Embrittlement of Metals by Trace Impurities 142

10 Ab initio Simulation of the Ni3Al-based Solid Solutions 145

10.1 Phases in Superalloys 145

10.2 Mean-Square Amplitudes of Atomic Vibrations in γ0-based Phases 147

10.3 Simulation of the Intermetallic Phases 148

10.4 Electron Density 154

11 The Tight-Binding Model and Embedded-Atom Potentials 157

11.1 The Tight-Binding Approximation 157

11.2 The Procedure of Calculations 160

11.3 Applications of the Tight-Binding Method 160

11.4 Environment-Dependent Tight-Binding Potential Models 162

11.5 Embedded-Atom Potentials 166

11.6 The Embedding Function 168

11.7 Interatomic Pair Potentials 170

12 Lattice Vibration: The Force Coefficients 175

12.1 Dispersion Curves and the Born–von Karman Constants 176

12.2 Fourier Transformation of Dispersion Curves: Interplanar Force Constants 181

12.3 Group Velocity of the Lattice Waves 183

12.4 Vibration Frequencies and the Total Energy 187

13 Transition Metals 193

13.1 Cohesive Energy 194

13.2 The Rectangular d Band Model of Cohesion 200

13.3 Electronic Structure 201

13.4 Crystal Structures 205

13.5 Binary Intermetallic Phases 205

13.6 Vibrational Contribution to Structure 211

14 Semiconductors 215

14.1 Strength and Fracture 219

14.2 Fracture Processes in Silicon 223

14.3 Graphene 224

14.4 Nanomaterials 227

15 Molecular and Ionic Crystals 233

15.1 Interaction of Dipoles: The van der Waals Bond 233

15.2 The Hydrogen Bond 236

15.3 Structure and Strength of Ice 239

15.4 Solid Noble Gases 242

15.5 Cohesive Energy Calculation for Noble Gas Solids 244

15.6 Organic Molecular Crystals 246

15.7 Molecule-Based Networks 248

15.8 Ionic Compounds 250

16 High-Temperature Creep 253

16.1 Experimental Data: Evolution of Structural Parameters 254

16.2 Physical Model 258

16.3 Equations to the Model 260

16.4 Comparison with the Experimental Data 261

17 Fatigue of Metals 263

17.1 Crack Initiation 264

17.2 Periods of Fatigue-Crack Propagation 267

17.3 Fatigue Failure at Atomic Level 270

17.4 Rupture of Interatomic Bonding at the Crack Tip 276

18 Modeling of Kinetic Processes 279

18.1 System of Differential Equations 279

18.2 Crack Propagation 280

18.3 Parameters to Be Studied 281

18.4 Results 282

Appendix A Table of Symbols 285

Appendix B Wave Packet and the Group and Phase Velocity 289

Appendix C Solution of Equations of the Kronig–Penney Model 291

Appendix D Calculation of the Elastic Moduli 293

Appendix E Vibrations of One-Dimensional Atomic Chain 295

References 299

Index 303